157 Boundary Operation
A mapping that maps a domain to its edge that is used to relate a definite integral over a domain to a definite integral on that edge.
Note: Also written \(\partial\). Also called boundary operator.
definition [d] (Boundary Operation = Boundary Operator = \(\partial\)) From Emam: in more advanced discussions, the symbol \(\partial\) is treated as an operator—a sort of derivative—and it has the property of being nilpotent, which means that applying it twice always vanishes:
- \(\partial^{2} = 0\) .
where
- \(\partial\) is the boundary operation.
- \(\partial M\) is the edge of a manifold or domain \(M\).
- \(\partial^{2} = 0\) means the edge of an edge is empty.
definition [d] (Boundary Operation) From Gowers: in Stokes’s theorem,
- \(\displaystyle \int_{S} d\omega = \int_{\partial S} \omega\)
for an oriented manifold \(S\) and form \(\omega\), where \(\partial S\) is the oriented boundary of \(S\). Differentiation \(\omega \mapsto d\omega\) is the adjoint of the boundary operation.
where
- \(\partial S\) is the edge of \(S\) under the boundary operation.
- \(d\omega\) is the exterior derivative of \(\omega\).
A mapping associates a function with another function. Under this mapping, the definite integral of a derivative over a domain has the same value as the definite integral of the function over the edge of the domain that is used to find a solution to an equation.
definition [d] (Boundary Operation) From Gowers, stated with elementary terms: an equation relates the definite integral of a derivative over a domain to the definite integral over its edge,
- \(\displaystyle \int_{D} Df = \int_{\partial D} f\) .
where
- both integrals are definite integrals.
- \(D\) is the domain, the set on which the definite integral of the derivative is taken.
- \(\partial D\) is the edge of the domain under the boundary operation.
- \(f\) is the function whose values appear in the definite integral on the edge.
- \(Df\) is the derivative of \(f\), which appears in the definite integral over the domain.
- \(\partial\) denotes the boundary operation.
157.1 Examples
Before the boundary operation acts, one has a domain set \(A\). After the boundary operation acts, one has an edge set \(B = \partial A\). Stokes’s theorem then relates a definite integral over \(A\) to a definite integral over \(B\).
example 1 [d] (Closed interval to endpoints — Nash and Sen; Emam) Before: take the domain set
- \(A = [a,b] = \{ x \in \mathbb{R} : a \le x \le b \}\) .
After: the boundary operation maps \(A\) to the edge set of endpoints
- \(B = \partial A = \{ a, b \}\) .
Nash and Sen record the same edge for the half-open interval: if \(U = [a,b)\), then \(U^{\circ} = (a,b)\) and \(\overline{U} = [a,b]\), so \(b(U) = \overline{U} - U^{\circ} = \{ a, b \}\). Emam states the same passage of sets in manifold language: if \(M = [a,b]\), then \(\partial M\) has dimension zero and is the pair of points \(\{ a, b \}\).
where
- \(A\) is the domain set before \(\partial\) acts.
- \(B = \partial A\) is the edge set after \(\partial\) acts.
- \(a\) and \(b\) are real numbers with \(a < b\).
example 2 [d] (Closed unit disk to circle — Lee) Before: take the domain set
- \(A = \overline{B}^{2} = \{ x \in \mathbb{R}^{2} : |x| \le 1 \}\) .
After: the boundary operation maps \(A\) to the unit circle
- \(B = \partial A = S^{1} = \{ x \in \mathbb{R}^{2} : |x| = 1 \}\) .
Lee: the closed unit disk \(\overline{B}^{2}\) is a manifold with boundary whose manifold boundary is the circle. Its topological boundary as a subset of \(\mathbb{R}^{2}\) is that same circle.
where
- \(A\) is the domain set before \(\partial\) acts.
- \(B = \partial A\) is the edge set after \(\partial\) acts.
- \(|x|\) is the Euclidean length of \(x\).
example 3 [d] (Upper half-space to hyperplane — Lee) Before: take the domain set
- \(A = \mathbb{H}^{n} = \{ (x_{1},\ldots,x_{n}) \in \mathbb{R}^{n} : x_{n} \ge 0 \}\) .
After: the boundary operation maps \(A\) to
- \(B = \partial A = \partial \mathbb{H}^{n} = \{ (x_{1},\ldots,x_{n}) \in \mathbb{R}^{n} : x_{n} = 0 \}\) .
where
- \(A\) is the domain set before \(\partial\) acts.
- \(B = \partial A\) is the edge set after \(\partial\) acts.
- \(n\) is a natural number with \(n > 0\).
example 4 [d] (Edge of an edge is empty — Emam; Nakahara) Apply \(\partial\) twice to a solid ball. Before the first application,
- \(A = D^{3}\)
is the solid ball. After one application,
- \(B = \partial A = S^{2}\)
is the sphere. After a second application the edge of that edge is empty:
- \(\partial B = \partial^{2} A = \emptyset\) .
Emam: the boundary of a boundary is always vanishing; a closed manifold such as a circle has empty boundary, written \(\partial M = 0\), and \(\partial^{2} = 0\). The same holds for a sphere. Nakahara: the boundary of the solid ball \(D^{3}\) is the sphere \(S^{2}\), and the boundary of the sphere is an empty set.
where
- \(A\) is the domain set before \(\partial\) acts.
- \(B = \partial A\) is the edge set after one application of \(\partial\).
- \(\partial^{2} A = \emptyset\) is the result after \(\partial\) acts on \(B\).
example 5 [d] (Before and after in Stokes’s theorem — Gowers) Before: a definite integral is taken over a domain set \(A = D\). After the boundary operation produces \(B = \partial D\), Stokes’s theorem moves that definite integral onto the edge set:
- \(\displaystyle \int_{A} d\omega = \int_{B} \omega = \int_{\partial D} \omega\) .
The sets alone change from \(A\) to \(B = \partial A\); the equality says the two definite integrals have the same value.
where
- \(A = D\) is the domain set before the move.
- \(B = \partial D\) is the edge set after \(\partial\) acts.
- both integrals are definite integrals.
157.2 Elementary Example
157.2.1 Simple
The boundary operation \(\partial\) maps a domain set \(A\) to its edge set \(B = \partial A\).
\[ A = \{ a,\ b,\ c,\ d \} \]
\[ B = \partial A = \{ a,\ b \} \]
where
- \(A\) is the domain set before \(\partial\) acts.
- \(B = \partial A\) is the edge set after \(\partial\) acts.
- \(\partial\) is the boundary operation.
157.2.2 General
Nilpotence says the edge of an edge is empty: \(\partial^{2} = 0\). A solid ball maps to a sphere, then to the empty set.
\[ A = D^{3} \]
\[ B = \partial A = S^{2} \]
\[ \partial B = \partial^{2} A = \emptyset \]
where
- \(D^{3}\) is the solid ball.
- \(S^{2}\) is the sphere, the edge of \(D^{3}\).
- \(\emptyset\) is the empty set.
157.3 References
- Emam, M. H. Covariant Physics. Oxford University Press, 2021. — \(\partial\) as a nilpotent operator; \(\partial^{2}=0\); line segment to endpoints.
- Gowers, T., Barrow-Green, J., & Leader, I. (eds.). The Princeton Companion to Mathematics. Princeton University Press, 2008. — boundary operation in Stokes’s theorem; adjoint of differentiation.
- Lee, J. M. Introduction to Topological Manifolds. Springer, 2011. — closed unit disk to circle; \(\mathbb{H}^{n}\) to \(\partial \mathbb{H}^{n}\).
- Nash, C., & Sen, S. Topology and Geometry for Physicists. Academic Press, 1983. — \(b(U) = \{ a, b \}\) for an interval.
- Nakahara, M. Geometry, Topology and Physics. Institute of Physics Publishing, 2003. — \(\partial D^{3} = S^{2}\) and \(\partial S^{2}\) empty.
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